REQUIRED RETURN ANALYSIS CAPITAL MARKET LINE (CML) THE CML

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RETURN CALCULATIONS

Required Return Analysis

Capital Market Line (CML)

The CML is an indicator of the trade-off between expected return and risk as measured by standard deviation for efficient portfolios.

Assumptions:

Homogeneous expectations
No transaction costs
All securities are infinitely divisible
No taxes
No trading price impacts, i.e., no individual impacts the market.
One period time horizon
Utilize the mean/variance criteria for decision making
Borrowing/Lending at the risk free rate

REQUIRED RETURN ANALYSIS CAPITAL MARKET LINE (CML) THE CML

Require Return = (Price of time) + (Amount of risk) (Price of risk)
Reward for waiting = price of time = y-intercept = R_Free
Reward per unit of Risk = the price of risk = slope = the market price of risk for all efficient portfolios (i.e., the extra return to gained by increasing the level of risk as defined by the standard deviation of the efficient portfolio by one unit).
Valid for efficient portfolios only.
By definition _RF = 0.

Note: R_M is the return on the market portfolio, usually proxied by some broad index of securities such as the S&P 500. Technically, the market portfolio should consist of an investment in all possible securities. Each security invested should correspond to its relative market value. The relative market value of a security is equal to the aggregate market value of the security divided by the sum of the aggregate market values of all securities.

Note: R_Free is the return on the risk-free security, usually proxied by some government security such as the T-bill.

Security Market Line (SML)

The SML is a graphical depiction of the CAPM

Capital Asset Pricing Model (CAPM)

The CAPM is an equation relating the required rate of return for any security (or portfolio) with the risk for that security as measured by beta.

Required rate of return_i = risk-free rate + risk premium
Reward for waiting = price of time = y-intercept = R_Free
Reward per unit of Risk = the price of risk = slope = the market risk premium

REQUIRED RETURN ANALYSIS CAPITAL MARKET LINE (CML) THE CML

Required rate of return_i = (Price of time) + (Amount of risk) (Price of risk)

where the Beta of Security_I equals

Market portfolio beta equals 1.
Risk-free security beta equals 0.

Portfolio beta: the weighted average of the betas on the securities that make up the portfolio (i.e., ).

Beta estimation: One method to estimate beta is to rely on historical returns. Regress the returns from the security against the returns on the market portfolio. This produces a characteristic line. The slope of this line is an estimate for the historical beta. Another alternative, which is more frequently used, is to use excess returns (i.e., the return from the security (and market) less the risk-free rate) rather than just returns.

Equilibrium Expected Returns: According to CAPM, asset prices will adjust until equilibrium occurs whereby each and every security plots on the SML.

E(R_ei) = R(R_i) = R_Free + [E(R_M) ‑ R_Free] _i

Note: E(R_ei) is the equilibrium expected return on security i.

Over-valued (over-priced) securities: When the expected return from a security is lower than the required return generated by the CAPM, then the security is said to be over-valued (over-priced). Since the market is somewhat efficient, there will be excess sell-side orders which will drive the price down to clear the market. This in turn will cause the expected return to increase.

Under-valued (under-priced) securities: When the expected return from a security is higher than the required return generated by the CAPM, then the security is said to be under-valued (under-priced). Since the market is somewhat efficient, there will be excess buy-side orders which will drive the price up to clear the market. This in turn will cause the expected return to decrease.

Separation Theorem: The idea that the decision of which portfolio of risky assets to hold is separate from the decision of how to allocate investable funds between the risk-free asset and the risky asset.

Arbitrage Pricing Theory (APT): An equilibrium theory of expected returns for securities involving few assumptions about investor preferences.

Total risk: Total risk of security i consists of market risk (inherent in all investments) and firm specific risk as defined by _i = [ _i² * ²_M + ²_e ]^0.5 where _i² * ²_M is the market risk (also called systematic risk and nondiversible risk) and ²_e is the firm specific risk (also called unsystematic risk and diversifiable risk)

Misprice Measure (Alpha): The difference between a security's expected return and an appropriate (equilibrium) expected return. For example with the CAPM, alpha is define as: _i = E(R_i) ‑ E(R_ei) = E(R_i) ‑ {R_Free + [E(R_M) ‑ R_Free] _i}.

Note: With the CAPM, a security's alpha is equal to the vertical distance by which it lies above or below the SML.

if _i > 0 then security i's expected return is above the SML and the security is underpriced;

if _i < 0 then security i's expected return is below the SML and the security is overpriced;

if _i = 0 then security i's expected return is on the SML and the security is correctly priced.

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